Wright functions governed by fractional directional derivatives and fractional advection diffusion equations

TL;DR

This paper explores fractional directional derivatives and fractional advection diffusion equations, deriving solutions in terms of Wright functions and discussing their probabilistic interpretations, advancing understanding of fractional PDEs.

Contribution

It introduces solutions to fractional advection equations using Wright functions and links fractional derivatives with stable densities, providing new analytical tools.

Findings

Solutions expressed via Wright functions

Connection established between fractional derivatives and stable densities

Probabilistic interpretations of fractional PDE solutions

Abstract

We consider fractional directional derivatives and establish some connection with stable densities. Solutions to advection equations involving fractional directional derivatives are presented and some properties investigated. In particular we obtain solutions written in terms of Wright functions by exploiting operational rules involving the shift operator. We also consider fractional advection diffusion equations involving fractional powers of the negative Laplace operator and directional derivatives of fractional order and discuss the probabilistic interpretations of solutions.